What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?
Let's get this equation of a circle into the form (x - h)2 + (y - k)2 = r2 , where (h, k) is the center of the circle, and r is its radius.
x2 + y2 - 7 = 4y - 14x + 3
Add 7 to both sides.
x2 + y2 = 4y - 14x + 10
Add 14x to both sides, and subtract 4y from both sides.
x2 + 14x + y2 - 4y = 10
Add 49 and add 4 to both sides to complete the squares.
x2 + 14x + 49 + y2 - 4y + 4 = 10 + 49 + 4
Factor x2 + 14x + 49 as a perfect square trinomial.
(x + 7)2 + y2 - 4y + 4 = 63
Factor y2 - 4y + 4 as a perfect square trinomial.
(x + 7)2 + (y - 2)2 = 63
Now that the equation is in this form, we know that..
r2 = 63
area of circle = pi * r2 = pi * 63 = 63pi